A class of remarkable submartingales

In this paper, we consider the special class of positive local submartingales ( X t ) of the form X t = N t + A t , where the measure ( d A t ) is carried by the set { t : X t = 0 } . We show that many examples of stochastic processes studied in the literature are in this class and propose a unified approach based on martingale techniques for studying them. In particular, we establish some martingale characterizations for these processes and compute explicitly some distributions involving the pair ( X t , A t ) . We also associate with X a solution to the Skorokhod’s stopping problem for probability measures on the positive half-line.